Cp and Cv

In-depth explanation of Cp and Cv

The heat capacity at constant pressure, Cp, is larger than the heat capacity at constant volume, Cv, because heating at constant pressure lets the gas expand and do work on the surroundings, so extra heat is needed. For one mole of an ideal gas the relation Cp minus Cv equals R holds exactly, where R is the gas constant of 8.314 joules per mole per kelvin. For monatomic ideal gases Cv equals 3/2 R, while for diatomic gases near room temperature Cv is about 5/2 R, reflecting rotational motion. These differences determine how gases respond to heating under different conditions. Relationships between Cp and Cv are essential in analyzing engines and adiabatic processes.

Going deeper: Cp and Cv

Key ideas: Remember Cp minus Cv equals R for one mole of an ideal gas, and use Cv for constant volume heating and Cp for constant pressure heating. Monatomic values are 3/2 R and 5/2 R, and diatomic values near room temperature are 5/2 R and 7/2 R.

Common mistakes and connections: Errors arise by swapping the two or applying the monatomic values to molecules. At high temperatures vibrations add to the heat capacity of diatomics. These constants set gamma and control adiabatic behavior and the speed of sound.

Visualization of Cp and Cv

Cp exceeds Cv by about R (8.31 J/mol/K) for gases because heating at constant pressure also pays for expansion work. The interactive lesson shows this as a live animation or chart.

Worked example and use case

Helium has Cp of about 20.8 and Cv of about 12.5 joules per mole per kelvin, and the difference is close to R.

Where Cp and Cv fits in Thermodynamics & Energetics

Cp and Cv is topic 52 of 91 in Thermodynamics & Energetics.

Related topics

• Adiabatic ProcessAdvanced
• Isobaric ProcessIntermediate
• Isochoric ProcessIntermediate
• Heat Capacity RatioAdvanced
• Carnot CycleAdvanced
• Heat Engine EfficiencyAdvanced

Browse all 91 topics in Thermodynamics & Energetics